An improvement on the bounds of Weil exponential sums over Galois rings with some applications

We present an upper bound for Weil-type exponential sums over Galois rings of characteristic p² which improves on the analog of the Weil-Carlitz-Uchiyama bound for Galois rings obtained by Kumar, Helleseth, and Calderbank (1995). A more refined bound, expressed in terms of genera of function fields, and an analog of McEliece´s (1971) theorem on the divisibility of the homogeneous weights of codewords in trace codes over Z_p², are also derived. These results lead to an improvement on the estimation of the minimum distance of certain trace codes over Z_p² and the bounds on the correlation of certain nonlinear p-ary sequences.